The analysis is set up to integrate the two halves of the problem starting from conjunction. It is important to have a good first guess for the initial conditions; i.e. the position and velocity of the spacecraft, Earth and Mars. Assuming the spacecraft is on a neighboring trajectory to a Hohmann Transfer not only allows these conditions to be established, but also greatly reduces computation time by puting the spacecraft on a flight path that is already near the optimal trajectory.
The position of Earth and Mars is established as the first conjunction after the J2000 epoch. This time is calculated directly from the J2000 ephemerides. The position of the spacecraft at this time is determined by assuming a two-body Hohmann Transfer between the orbits of Earth and Mars. The time for Earth to move from periapse of the Hohmann to Earth-Mars conjunction is calculated, and the position of Earth at the beginning of the Hohmann Transfer is found. The spacecraft starts at this same position, and the spacecraft's position at Earth-Mars conjunction is determined by integrating it along the two-body Hohmann Transfer the same increment of time. Typically, the spacecraft is 5 to 10 degrees behind Earth, at an approximate true anomaly of 90 to 93 degrees.
When the user specifies an extra thrust to be applied at Earth, this forces a change in the Hohmann Transfer ellipse. The same initial position of the ellipse (i.e. periapse) is used, but the apoapse is greater because of the higher thrust applied at Earth. The geometry of this new ellipse is calculated, and the initial position of the spacecraft on this ellipse is calculated as before.
